All of the 3rd grade teachers and students from Covington went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$4.50$ each for students, and the group paid $$66.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$26.00$ each for teachers and $$12.00$ each for students, and the group paid $$212.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+4.5y = 66.5}$ ${26x+12y = 212}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-26x-18y = -266}$ ${26x+12y = 212}$ Add the top and bottom equations together. $ -6y = -54 $ $ y = \dfrac{-54}{-6}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {6.5x+4.5y = 66.5}$ to find $x$ ${6.5x + 4.5}{(9)}{= 66.5}$ $6.5x+40.5 = 66.5$ $6.5x = 26$ $x = \dfrac{26}{6.5}$ ${x = 4}$ You can also plug ${y = 9}$ into $ {26x+12y = 212}$ and get the same answer for $x$ ${26x + 12}{(9)}{= 212}$ ${x = 4}$ There were $4$ teachers and $9$ students on the field trips.